Workshop on Representations, Support, and Cohomology – Abstracts
Paul Balmer (Los Angeles)
The spectrum of the equivariant stable homotopy category
Every tensor-triangulated category admits a geometry, which begins with a topological space called its "spectrum". We'll review some of the basic results in this direction for the usual stable homotopy category. Then, we'll turn to the equivariant version, which is morally "the most fundamental equivariant tensor-triangulated category" one can associate to a finite group. The latter is joint with Beren Sanders.
Stephen Donkin (York)
Polynomially and infinitesimally injective modules
This is joint work with Haralampos Geranios. The support varieties of Weyl modules have received a lot of attention in recent years. We start by reviewing this. We then make some remarks on support varieties of tilting modules. We then concentrate on the related basic problem of determining which polynomially injective modules are injective as restricted modules for the Lie algebra of the general linear group. We relate this to the problem of determining the composition factors of an m-fold tensor product of the symmetric algebra of the natural module for the general linear group and explain how far we have got with this problem.
Christopher Drupieski (Chicago)
Support varieties for Lie superalgebras and finite graded group schemes
Following the pioneering work of Quillen in the 1970s, Carlson, Avrunin and Scott, Friedlander and Parshall, Jantzen, and others made much progress in the 1980s studying the cohomology and representation theory of finite groups and restricted Lie algebras by way of their associated cohomological support varieties. Later, many of these methods and results were generalized first to infinitesimal group schemes by Suslin, Friedlander, and Bendel, and then to arbitrary finite group schemes by Friedlander and Pevtsova. In this talk I will discuss some results and conjectures concerning how some of the aforementioned methods and results can be generalized to restricted (and non-restricted) finite-dimensional Lie superalgebras and to certain finite graded group schemes. This is joint work with Jonathan Kujawa.
Karin Erdmann (Oxford)
Ext-finite and non-periodic bounded modules for selfinjective algebras
We consider finite-dimensional selfinjective algebras over a field. To have a theory of support varieties for modules, one would like that the finite generation conditions known as (Fg) for Hochschild cohomology hold. A module over this algebra is said to be ext-finite if its ext-algebra is finite-dimensional. We call it a 'criminal' if it has a bounded projective resolution but is not periodic. It is known that if (Fg) holds for the algebra then ext-finite modules are projective, and the algebra does not have criminals. We discuss the existence of non-projective ext-finite modules, and of criminals, for wekly symmetric special biserial algebras.
Jens Carsten Jantzen (Aarhus)
Restrictions from 𝔤𝔩n to 𝔰𝔩n
Let L denote the general linear Lie algebra of all n-by-n matrices over an algebraically closed field F, and let L' be the Lie subalgebra of L consisting of all matrices with trace 0 (the special linear Lie algebra). If the characteristic of F does not divide n, then any finite dimensional simple L-module restricts to a simple L'-module. This is not true when the characteristic divides n, and I shall show precisely (with the help of two of Rolf Farnsteiner's results) which simple modules become reducible under restriction.
Paul Levy (Lancaster)
Nilpotent commuting varieties and support varieties
Commuting varieties are classical objects of study in Lie theory. Historically the first result was the proof by Motzkin-Taussky in 1955 (established independently by Gerstenhaber) that the variety of pairs of commuting n×n matrices is irreducible. Richardson extended this to an arbitrary reductive Lie algebra in characteristic zero. More recently there has been interest in the subvariety of pairs of commuting nilpotent elements. One highlight was Premet's proof that this nilpotent commuting variety of a reductive Lie algebra is equidimensional, and is irreducible for 𝔤𝔩n. In positive characteristic, the nilpotent commuting variety is related to cohomology of the second Frobenius kernel of the corresponding group by a result of Suslin-Friedlander-Bendel.
Motivated by this connection, I will explore two variations on the theme. First of all, I will introduce some generalisations of the nilpotent commuting variety, for which we confine the first coordinate to a fixed nilpotent orbit closure. The main task here is to determine the irreducible components and their dimensions. This is joint work with N. Ngo. Secondly, I will summarize some recent results on the variety of r-tuples of commuting nilpotent elements of 𝔤𝔩n or 𝔰𝔭2n. This is joint work with N. Ngo and K. Šivic.
Julia Pevtsova (Seattle)
π-points, cosupport, and stratification
The problem of classifying thick subcategories in a given triangulated category goes back to the seminal work of Devinatz-Hopkins-Smith in stable homotopy theory and Hopkins, Neeman, and Thomason in algebraic geometry. I'll describe some recent progress on the question of classification of tensor ideals in a stable module category of a finite group scheme, and how it relates to π-points and π-support and cosupport. This is joint work with D. Benson, S. Iyengar and H. Krause.
Gerhard Röhrle (Bochum)
Cocharacter-closure and the rational Hilbert-Mumford Theorem
I shall introduce the notion of cocharacter-closure and how this leads to a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We will illustrate with some examples how this concept differs from the usual Zariski-closure and discuss some applications. This reports on joint work with M. Bate, S. Herpel and B. Martin.
Øystein Skartsæterhagen (Trondheim)
Relations between algebras preserving the (Fg) condition for support varieties
The theory of support varieties for finite-dimensional algebras was introduced by Snashall and Solberg in 2004, and is formulated in terms of the Hochschild cohomology ring. To ensure that the support variety theory for a given algebra has good properties, the algebra and its Hochschild cohomology should satisfy the finite-generation condition (Fg), defined by Erdmann, Holloway, Snashall, Solberg and Taillefer.
In this talk, I will present three results which show that certain relations between algebras preserve the (Fg) condition:
- Let A be an algebra and let e be an idempotent of A satisfying certain assumptions. Then the algebra A satisfies the (Fg) condition if and only if the algebra eAe does.
- The (Fg) condition is preserved under derived equivalence of algebras.
- The (Fg) condition is preserved under singular equivalence of Morita type (with level) between Gorenstein algebras.
Result (1) is from a joint work with Chrysostomos Psaroudakis and Øyvind Solberg. Result (2) is from a joint work with Julian Külshammer and Chrysostomos Psaroudakis.
Oded Yacobi (Sydney)
Lusztig slices and truncated shifted Yangians
Lusztig slices are transverse slices to affine Schubert varieties in the affine Grassmannian of a reductive group G, which arise naturally in the geometric Satake correspondence. We will discuss algebras called truncated shifted Yangians, which are quantizations of these slices. In type A, these algebras generalize finite W-algebras. We will describe the highest weight theory of these algebras using Nakajima's monomial crystal. This leads to conjectures about categorical actions of the Langlands dual Lie algebra of G on certain representation categories of truncated shifted Yangians.